Abstract

The lowest-order Gaussian beam mode is considered in a high-aperture theory based on a new variation of the complex source point model. To avoid singularities, combinations of sources and sinks are assumed. The resultant beam is a rigorous solution of Maxwell’s equations for all space that reduces to the conventional Gaussian beam in the paraxial limit and that is physically realizable. The field in the region of the waist and far from the waist is explored. It is demonstrated that direct rigorous evaluation of the field is feasible.

© 1999 Optical Society of America

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References

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  1. C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
    [CrossRef]
  2. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  3. S. Y. Shin, L. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [CrossRef]
  4. A. E. Siegman, “Hermite–Gaussian functions of complex arguments as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  5. M. Couture, P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [CrossRef]
  6. A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
    [CrossRef]
  7. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  8. J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  9. A. G. van Nie, “Rigorous calculation of the electromagnetic field of a wave beam,” Philips Res. Rep. 19, 378–394 (1964).
  10. A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–291 (1974).
  11. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  12. M. Lax, W. H. Louisel, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  13. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  14. G. P. Agrawal, D. N. Pattanayck, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  15. L. W. Davis, G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6, 22–23 (1981).
    [CrossRef] [PubMed]
  16. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  17. L. W. Davis, G. Patsakos, “Comment on representation of vector electromagnetic beams,” Phys. Rev. A 26, 3702–3703 (1982).
    [CrossRef]
  18. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  19. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [CrossRef]
  20. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).
    [CrossRef]
  21. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
    [CrossRef] [PubMed]
  22. P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
    [CrossRef] [PubMed]
  23. P. Varga, P. Török, “Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave equation,” Opt. Commun. 152, 1–3 (1998).
    [CrossRef]
  24. C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
    [CrossRef]
  25. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  26. C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
    [CrossRef]
  27. C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
    [CrossRef]
  28. J. J. Stamnes, V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Phys. 5, 195–226 (1996).
  29. C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).
  30. C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
    [CrossRef]
  31. M. A. Porras, “The best optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
    [CrossRef]
  32. X. D. Zeng, C. H. Liang, Y. Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 2042–2047 (1997).
    [CrossRef] [PubMed]

1998 (2)

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

P. Varga, P. Török, “Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave equation,” Opt. Commun. 152, 1–3 (1998).
[CrossRef]

1997 (3)

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).

X. D. Zeng, C. H. Liang, Y. Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 2042–2047 (1997).
[CrossRef] [PubMed]

1996 (2)

J. J. Stamnes, V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Phys. 5, 195–226 (1996).

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[CrossRef] [PubMed]

1994 (2)

C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

M. A. Porras, “The best optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

1987 (2)

1986 (2)

1985 (1)

1983 (1)

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1982 (1)

L. W. Davis, G. Patsakos, “Comment on representation of vector electromagnetic beams,” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

1981 (2)

L. W. Davis, G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6, 22–23 (1981).
[CrossRef] [PubMed]

M. Couture, P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

1979 (3)

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

G. P. Agrawal, D. N. Pattanayck, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

1978 (2)

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

1977 (1)

1975 (1)

M. Lax, W. H. Louisel, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1974 (2)

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–291 (1974).

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

1973 (1)

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1964 (1)

A. G. van Nie, “Rigorous calculation of the electromagnetic field of a wave beam,” Philips Res. Rep. 19, 378–394 (1964).

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

G. P. Agrawal, D. N. Pattanayck, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

An, Y. Y.

Asakura, T.

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–291 (1974).

Bélanger, P.-A.

M. Couture, P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Berry, M.

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Couture, M.

M. Couture, P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Cullen, A. L.

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

Davis, L. W.

L. W. Davis, G. Patsakos, “Comment on representation of vector electromagnetic beams,” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

L. W. Davis, G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6, 22–23 (1981).
[CrossRef] [PubMed]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Dhayalan, V.

J. J. Stamnes, V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Phys. 5, 195–226 (1996).

Felsen, L.

Fukumitsu, O.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Larkin, K. G.

C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisel, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Liang, C. H.

Louisel, W. H.

M. Lax, W. H. Louisel, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Matthews, H. J.

McKnight, W. B.

M. Lax, W. H. Louisel, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mukunda, N.

Nye, J. F.

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Patsakos, G.

L. W. Davis, G. Patsakos, “Comment on representation of vector electromagnetic beams,” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

L. W. Davis, G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6, 22–23 (1981).
[CrossRef] [PubMed]

Pattanayck, D. N.

Porras, M. A.

M. A. Porras, “The best optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Saghafi, S.

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).

C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

Simon, R.

Stamnes, J. J.

J. J. Stamnes, V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Phys. 5, 195–226 (1996).

Sudarshan, E. C. G.

Takenaka, T.

Török, P.

P. Varga, P. Török, “Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave equation,” Opt. Commun. 152, 1–3 (1998).
[CrossRef]

C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[CrossRef] [PubMed]

van Nie, A. G.

A. G. van Nie, “Rigorous calculation of the electromagnetic field of a wave beam,” Philips Res. Rep. 19, 378–394 (1964).

Varga, P.

P. Varga, P. Török, “Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave equation,” Opt. Commun. 152, 1–3 (1998).
[CrossRef]

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[CrossRef] [PubMed]

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Yokota, M.

Yoshida, A.

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–291 (1974).

Yu, P. K.

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

Zauderer, E.

Zeng, X. D.

Appl. Opt. (2)

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

IEE J. Microwaves, Opt. Acoust. (2)

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

J. Mod. Opt. (2)

C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

P. Varga, P. Török, “Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave equation,” Opt. Commun. 152, 1–3 (1998).
[CrossRef]

M. A. Porras, “The best optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (2)

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–291 (1974).

C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).

Philips Res. Rep. (1)

A. G. van Nie, “Rigorous calculation of the electromagnetic field of a wave beam,” Philips Res. Rep. 19, 378–394 (1964).

Phys. Rev. A (6)

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Couture, P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

M. Lax, W. H. Louisel, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

L. W. Davis, G. Patsakos, “Comment on representation of vector electromagnetic beams,” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. R. Soc. London, Ser. A (3)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Pure Appl. Phys. (1)

J. J. Stamnes, V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Phys. 5, 195–226 (1996).

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Figures (8)

Fig. 1
Fig. 1

Variation in time-averaged electric energy density, normalized to unity at the focus, for the LP01 beam along the three axes for different values of kz0. Dashed curves, (sin kr/kr)2.

Fig. 2
Fig. 2

Time-averaged electric energy density, normalized to unity at the focus, in the waist for the LP01 beam for kz0=2.

Fig. 3
Fig. 3

Transverse variation in time-averaged total energy density, normalized to unity at the focus, for different values of kz0.

Fig. 4
Fig. 4

Axial variation in phase for the LP01 beam for different values of kz0, showing the Gouy phase anomaly.

Fig. 5
Fig. 5

Contours of constant time-averaged electric energy density, normalized to unity at the focus, in meridional planes, for kz0=2.

Fig. 6
Fig. 6

Variation in the three components of the electric field strength in the waist for the LP01 beam for kz0=2: (a) Ex, (b) Ey, (c) iEz.

Fig. 7
Fig. 7

Radiation pattern showing the normalized far-field intensity for different values of kz0.

Fig. 8
Fig. 8

Variation in the angle at which the far-field intensity drops to one half of the on-axis value for different values of kz0.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

f=w02z0=1kw0=12kz0,
z0=kw02/2.
f(kr)=j0(kr)+j2(kr)=-3cos kr(kr)2-sin kr(kr)3,
g(kr)=j0(kr)-12j2(kr)
=32sin krkr+cos kr(kr)2-sin kr(kr)3,
E=g(kr)+[f(kr)-g(kr)]x2r2+i2f(kr)kzi+[f (kr)-g(kr)]xyr2j+[f (kr)-g(kr)]xzr2-i2f (kr)kxk,
R=[x2+y2+(z-iz0)2]1/2,
WE=2E2
WT=2[f2(kR)+g2(kR)]+8(kR)2f2(kR),
E=[g(kR)+ikR f (kR)/2]i,
Φ=Arg{g[k(z-iz0)]+ik(z-iz0) f[k(z-iz0)]/2},
E=-3i4krexp(ikr)exp(kz0 cos θ)[(1+cos θ-sin2 θ cos2 ϕ)i-sin2 θ sin ϕ cos ϕj-sin θ(1+cos θ)cos ϕk],
E=-3i4krexp(ikr)exp(kz0)×exp[-kz0(1-cos θ)](1+cos θ).
a(θ)=1+cos θ2exp[-kz0(1-cos θ)]=cos2θ2exp-2kz0 sin2θ2.
Ex=1-f22[1+(iz)/z0]-5(x/w0)2+3(y/w0)2[1+(iz)/z0]2+[(x/w0)2+(y/w0)2]2[1+(iz)/z0]3ψ,
Ey=2f2(x/w0)(y/w0)[1+(iz)/z0]2ψ,
Ez=-2if(x/w0)[1+(iz)/z0]ψ,
ψ=1[1+(iz)/z0]exp-(x/w0)2+(y/w0)2[1+(iz)/z0]exp(ikz).

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